General Topology sncl its Applications 10 (1979) 147-160 @ North-Holland Publishing Company

R. LOWEN

Deparlnrent of Mathematics, Vtije UI:iversitcit Brussel, Brussel10.50, Belgium

Received 6 June 1977

In the first paragraph we study filters in the lattice Px, where I is the unitinterval and X an arbitrxy set. The mai t result of this section is a characterization of mr!:if;laI prime filters in Ix containing a given filt :r in Ix by means of ultrafilters on X.

In the second paragraph we apply the results of the previous section to define convergence in a fuzzy topological space which enables us to characterize fuzzy compactrless and fuzzy continuity.

AMS Subj. Class.: Primary 54,.22; Secondary 54.18

fiket-s prime-filters

fuzzy convergence fuzzy compactness fuzzy continuity

Introduction

Ix partially ordered by fi s P if’! p(x) s V(X) for each x E X is zg Brouwerian lattice and - I being a complete chain - it is also completely distributive [I].

Yet since it is not a Boolean algebra (it is not complemented) not all prime filters in Ix are maximal [2]. Moreover maxima) filters are not sufficient to describe all filters since Ix is not separative.

Therefore we introduce minimal prime filters containing a given filter and we give a characterization of these minimal prime filters using ultrafilters on X

1. Prelimiuaries

The notion of a filter in a lattice (see [2]) leads to the following definitions.

efinitioa 1.1. A subset 8 c Ix is a prefiher iff $J # E4 and (i) forall~,u~~wehave~hVEb; (ii) ifE.c~vandv@,then~@;

(iii) Ok%.

147

148 R. Lawen / Convergence in fuzzy topological spaces

2. A subset % c Ix is a base a prefilter iff +@ Z 0 and ~,v~@thereisa~~&such

(ii) O& 6% The prefiltel- 8 generated by (5) is defined as

and is denoted by (a). A subset @ of 8 is a base for !$ iff for all h ~3 there is a UC@ such that

j,& 2 Ye

.3. A subset $j c ix is a geraerating family or subbase for a prefilter iff the family of finite lower bounds of members of @ is a base for this prefilter. The prefilter generated by a subbase $j is also denoted by (6).

A prefilter 3 is called a prime prefilter iff for all EC, u E Ix such that ~vyc$jwehaveeither&‘jorvE&

For any g E 1*C; is the prsfilter generated by the singleton {JL}, i.e.

For any number c( E 1 and any x E X we designate axx E Ix as the function defined by axx(y ) = 0 if y # x and ax&) = a. It is clear that fi is prime iff p = uxx for some a#OandxCK

For two prefilters 8 and @ such that 8 c (8 we shall say that 3 is coarser than 69 and that @ is finer than 8.

The infinum and supremum of a family of prefilters is defined in the same way as for filters on X

As far as notation is concerned prefilters will be denoted by Gothic letters, filters on X by script letters and functions from X to I by lower case Greek letters.

If 8 is a prefilter and p E I* we shall want to know “how much 8 is alJove cc”. TO make this precise we introduce the following subset of I

and we call this subset the charucteristic set of ;% wirh respect to p. The number sup W(s) will be called the characteristic value of 3 with respect LO /IC and we denote it by P(z).

A set B is a characteristic set iff is one of the following 0, {O}, [O, c[ for any c E I, or [e, c] for any c E 1\(l).

R. Lowen / Convergence in fuzzy topologicar spaces 149

We shajll adapt the following notation

W(X) = {prefilters},

Wg(X) = (8 E W(X): u”(g) = K}

where K is some nonempty characteristic set. A member of WXX) will be called a K-filter.

n 2.1. Let fl E Ix. Then (i) if 2’ is a family of prefilters we have

(ii) if Z is a family of prefilten such that LJ 8Ea ‘3 is a subbase far some pre_%ter, then we have

ProoL (i) For all ($5 E 3, (7aEz 3 c @5 50 that clearly

v?p[ r‘la) = J u”(g). 8E9)5 &z9!

Suppose conversely that a E I\IJ gEtP: %Y@), then for all 3 E 2’ there exists u G 3 such that~~(~+a)~landconsec~uentlyforall~~~~,(~+a)~l~~or((ul.+~)~l~~ &‘& 3. Since obviously V(r;‘(p + 4) A 1)) C= [0, a[ we have a E %‘@(&s $?j*

(ii) The verification is trivid. That we have no more is shown by the following counter-example. Let X = I let JJ be the identity mapping and let Y = 1 -cc, then %?“(k) = @O(G) = [O, l[ but %O((k v ti)) = %“((& i\ V)) = [O, $[ cl

In what follows we shall mainly deal with the characteristic set and value of a prefilter with respect to 0.

We shall then simply speak of characteristic set and value and we shall omit the superscript 0 in our notation.

IF(X) will denote the set of all filters on X. Let K be a nonempty characteristic set, then we define ,the following mappings

@K :ff(X)+!N&X),

LK: u wK’(X)-*~(X), K-K

+{~L’]k, 11: p E 8, k dC}.

The verification that these definitions make sense and the proof of the following proposition are straightfo~ard and are left to the reader.

.2, Let K and Ii’ be characteristiti sets such that K’ c K, then for adi

i”urthermore the muppings WI;’ and &li: are ~~de~rese~ing.

Using analogous techniques as in set theory one can easily show that WHX) is inductive. Zorn’s lemma then yields the existence of maximal elements, which we shall call maximal K-filters. l

Now if $‘j is a maximal K4lter then since OK 0 ck: (8) is also a K-filter it follows from Proposition 2.2.(iv) that 8 = WK 0 l~c.(g).

core 2. [# % is an ultra#lter on X, then OK (1111) is a maximal K-filter, and if 8 is a prime orefilter then fur all ~hu~~~te~istic sets K c %T@), b&j) is an ultp~~lter on X.

roof. Let % be an ultrafilter on X and let 8 E WK (X) be a maximal K-filter, finer than OK(%). Then LX(~) =) CR 0 UK(%) = $$ and thus Q&) = %. Since 8 is maximal we have 8 = @k: 0 s&) = ok:(%).

As for the second statement let 8 be a prime prefilter and let A CK Then

,YAVXA~= 1 E 8 and so for instance XA E 8. Since always OE K we have A = x;;’ 10, l] E hk: (8) which shows that ~(8) is an ultrafilter on X. Cl

If 3 is a rn~~irnu~ K-fiber, then it is a prime Brewster,

If 8 E WK (X) is maximal then Q(‘) is ultra. Indeed, let % be an ultrafilter on X finkr than LK(~) then ~&Y)~w K 0 L&)= 8 and since OK(%) is a K-filter we have OK(%) = 8. This in turn implies that % := LK 0 OK (%) = CK (g).

Now Bet g, v ~1~ be such that p v v E 8, then for all k E we have (M -‘]k, 1] E I#@).

is of the form [O, ko] it suffices to remark t at for instance and since for all ire G k. p-‘]ko, l] c ~?]k, 11, we have H-*]k, kf

R. towen / Convergence in fuzzy topologicd spaces

If K is of the form [O, kO[ we choose a squence (kn)nsN of increasing nurirb- 5 in K

such that sup neM k, = ko. Then for all n EN either h-‘]k,, 1) E ~&$j) or /]& I] 1~

L&). Thus there exists a subsequence (k’ ) n neN of &.&N such that for # ;z E~V for instance ~-‘]k;, l] E L&).

ForanykEKchoosen~Nsuchthatk4&,thenagain~-’]k, l]~~-‘]~~..+ 1]aa3 thus p-‘]k, l] E ~~(3). Consequently in either case we have & E UK 0 &K(g) == $5 which proves that 3 is a prime prefilter. I3

The following corollary is an immediate consequence of the previous tl~.~rcr.~~

Corollary 2.5. If % 3s an ultrafilter on X then for all nonempty characteris& SC..:: K, OK(%) is a prime prefilter:

3. Structure of ptie prefdters

Remark. In analogy with filters on X one might think that every K-filter is equal *:o the intersection ofathe family of maximal K-filters which are finer. Yet this is qot the case as is shown by the following counterexample. Let X be arbitrary u:nd let 3 E !#&(X) such that ;a # cr)K 0 &K(g) = a. Further put

g(g) = (6 : $j maximal K-filter finer than g},

%?(@) = (6 : 6 maximal K-filter finer than (3)).

Clearly a(s) 1 a(@). But if $j E a(g), then @ = OK O &K ($) = UK O 6K (8) ‘.= @ so that 6 E %(a). Consequently a@) = .%!(a).

Let now P(g) be the set of all prime prefilters finer than 8 E W(X). From. [a]

But as we shall see further on 9(g) is too large a set for our purposes. The following proposition shows that we can extract a subset of P(g) wR!ch still

contains all the relevant information.

ropositbm 3.1. The set P(g) is inductive in the sense that every descending chniat of prefilters in it has a lower bound.

roof. Let 9 c S(z) be a descending chain. Consider ($50 = ~~SJ=B @. Then cle:u+l~~ @o 2 3. Next, if p v u E @o, then either for all @ E $3 we Ihave g E @ and thus p E @tir and we are done, or there exists some & E SR such that ,~,$@. Then since p w or E @ the latter implies that Y E $ for all @ E 3 such that 6 =) a. And if @ E such that @ c a, then since p# @ we must again have v E 6. iz

It now follows from Zorn’s rtheorem that there exist mmimal elements in shall denote the family of minimal elements in B(B) by !P,@),

152 R. Lowen / Convergence ilr fuzzy tu~~gjcul sptzces

It then also follows at once that we still have

In order to characterize these minimal prime prefilters in 9&‘$) in a more tangible way we need the following concept.

A filter S on X and a prefilter 8 are said to be compatible iff for all FE 9 and p E ‘& p does not vanish everywhere on E

We shall use the notation

~F:X++I:X+&~) if xeF,

Then since for all p, Y E 8 and F, G E S we have VZ; h FG = (v /\ &,G it is clear that if $s; and 3 are compatible then

is a Iprefilter.

3.2. Let 8 be a prefilter. Then

P,,, (8) = (@, %): 4% ultra on X and compatible with 8).

af. Let % be an ultrafilter on X compatible with 3. To show that (3, %) is prime let a, p E Ix such that cy v p E (3, %) then there exist JL E 8 and U E 12d such that cu~/Sz~~.Let thenA={x:~(x)~~~(x)}and~={x:~(x)~~~(x)].

Since A u B = X we have for instance A E %. Then since tu * VAnU this implies that r~ E (8, a). consequently (8, a) is prime. To show that it is minimal let @J E S(g) be such that (8, a) =3 (3 =) 8 and suppose that there exists p E 8 and U’ E % such that pu& 6

Then since p E @ and @ is prime we must have PURE ($5 and thus PUCE @,4k) which is impossible. Consequently (8, %) is minimal and thus

(8 Q)E %S@)* To show the converse let (8 E Pm@). Consider the characteristic set K = U(@) of

@. It follows from Theorem 2 3 that ~~(~~ is an ultra~lter on X And it is trivial that 8 is compatible with Q&Q. Now since 9$,,(@5) = {a} it follows from the first part of the theorem that (@, ~~(~)) = @. Since $j c @ it folio-;rvs that

Finally since (3, LK (a)) is prime we have ($5 = (5, LK $3)). u

R. Lmw~ / Convergence in fuzzy topological spaces 153

se i

Now let be a fuzzy topological space (fts for short) ([3]9.

. If g is a prefilter, then we define the adherence of 3

adh B(x)= inf F(x) UE%

where v’ is the fuzzy topokgical closure of Y [3].

efinition 4.2. If 8 is a prefilter, then we’define the limit of 8

lim B(x) = ink: adh a. GkSP,iA)

If 8(R) had been used in Definition 4.2 instead of 9,(a) then the limit of any prefilter in any fts would have been zero..

As an immediate consetluence of these definitions we have the following result, the proof of which is left to t

Proposition 4.1. Let 3 aqd @ be prefilters. (i) if 3 =) a, then adh 8 s adh a,

(ii) lim 3 G adh 3, (iii) if 3 is prime, then lim 3 = adh 3.

There 1s no relation between the limits of comparable prefilters as will be shown after Theorem 4.2.

If X is a topological space with topology F.. then in a natural way ‘we can associate a fts with it [3] where the fuzzy topology on X is ~($9, the family of lowee semcontinuous mappinlgs from X to the unit interval. Given a filter 9 and a characteristic set K o!n X we can compute adh OK($) and lim @JK($). In what follows if F c X, then 1? will denote the topological clc)s;ure of F.

.2. In the fits (X, o(T)) we have that (i) adh m&&q = (suj:, K) g XadhS,

(ii) hn UK (9) = (sup K) l Xlim SF*

Proof. (i) It is an imme91iate consequence of the definition of lower semicontinuous functions that for all Q r:: I and F c X zF= axF. Now let F E 9 and a E I\K, then obviously axe E OK($) so that

(SUP K) l Xadfl 5: = inf (axa& b) ad\K

154 R. Lowen / Convergence in fury topological spaces

Conversely suppose there exists x E adh 9 and Y E OK(@) such that F(x)< sup K. Then for E > 0 such that Z(u)+ e < sup K there exists a neighborhood V of x such that for all y E V 5(y)<i+)+s<supKButthenG-‘Cji;(x)+r,l])nV=0w is a contradiction.

(ii) We first need to show that for all 8 E 9,&&F)) V(@) = K. K 3 %‘(a) is clear. To show the remaining inclusion first remark that an ultrafilter % is compatible with W&F) iff all 3 5J-F Thus there is an ultrafilter % finer than 9 such that @ = (O&S), %). Hf k E K, then for all v E ~~(9) v-‘Ok, 11) E P= % so that for all U E %, ~?(]k, l]) n U T 0. Consequently for all v E 8 there is x E X such that V(X) > k, which implies that k E %(CS).

We can now apply &K to @ and Theorem 2.3, then states that CJ@) is an ultrafilter and next that w&J%)) is a maximal K-filter finer than a. Moreover LK (a) 3 &&w&&q) = @. This means that for all @ E g,,&&)) there exists % = LK (a) finer than 9 such that OK(%) 3 @ and so we have

= (SUP IL)- Xlim S-

Now since lim wK (9) G adh oK (-g) = (Sup K) l Xadh 9 it iS dear that for all x E lim 9 we have

lim ~K(~J(X)= (SUP K)Xlim s(x)*

Now let x & lim .F. Then there exists an ultrafilter a2rr finer than 9 and U E %? such that x& fi Since w&Q) is prime there exists 650~ P&K(~)) such that UK(%) 3 @I&-, 3 w&F). Clearly xu E o&U). Suppose that xv 6 80 then since &O is prime xv= E a0 and consequently xv= E OK(%) which is impossible. Thus xv E (5.50 and we have

The following counterexample now shows that there need be no relation between the limits of comparable prefilters.

Let (X, r) be a non-Hausdorff topologica! space and let 9 and $9 be filters such that 9 is strictly finer than %, lim ?I # 0, and lim g 3 lim 9%

Further let M’ 3 K be characteristic sets. Then although WK (9) is finer than OK#( 3) it follows at once that lim OK(%) and lim OK, W&Z?) are incomparable.

actnew

The use of characteristic sets of prefilters with respect to fuzzy sets becomes fully apparent when we want to give necessary and sufficient conditions for a fts to be fuzzy compact. ([3])

holds. ‘ The fts X k,kzzy compact ijj$ne of the following equivalent conditions

I?. Lowen / Convergence in fuzzy topological spaces

(i) For each constmt fuzzy set cy E Ix (a f 1) and for each prefilrrer 8 E W”,(R) there exists an x E X sawh that

(ii) For ?ach constmt fuzzy set a! E Ix (a f 1) and for each prime prefilter 8 E W:(X) there exists a.!: x E X such that

lim g(x) > Q,

(iii) For each prefii’w 8 E W+(X)

sup adh 3(x 1 .a c(B).

XC%

(iv) For each prime prefilter 8 E W(X)

sup lim 8(x 3 3 c(g). XEX

Proof. If X is fuzz.y compact, then let @M/Q+(X), put ‘23 = (2 u E 3) and choose E E V”(g)\(O). Thefdp Ior all finite subfamily B0 of 23 there exists an or’ E X such, that inf VEp)O V(X) > a! + E. Consequently it follows from the definition of fuzzy compnct- ness that there exists an x E X such that

Conversely to show rhat (i) implies fuzzy compactness let Iii)’ be a family of closed fuzzy sets such thalr for some 8 > 0 anti for all finite subfamily !%$“I c 23’ there iis an x E X such that

inf u(x)> C’Y +E. u&3&

Then consider the kamily

8l f = in ’ v: mh finite subset of 23’ u&, I

Clearly 8 generatc q a prefilter i! in WV”,@ ), and thus it follows fl-om (i): that there exists an x E X suctl that

inf v(x)== adh g(x)>cw Y Em’

which means that Jc is fuzzy compact. That (i) implies (ii) is trivial. To show the converse let 3 E V+ j:(X), then since

156 R, Lowen / Convergence in fuzzy topological jrpaces

it follows from proposition 2.1 that there exists 01 E ~~(~)n~~~). Then adh 8 2 adh @ = lim @ and consequently it follows from (ii) that there is an x E X such that adh fF(x)>ar.

We shall still! show that (ii) is equivalent to (iv). The equivalence of (i) and (iii) goes exactly the same. §uppose (ii) holds, then take a prime pr&lter &S clearly we have that for all Q! < c(g), 3 E VW”,(X) and consequently for all cy < c(B) there is xar E X such that lim 3(x,)> cy, thus

sup lint Q(x) 2 u$tTB, Q! = c (6) XEX

Conversely suppose (iv) holds, then let ar E Ix be constant and let 3 E VW?(X), this means that %” (3) 3 {O} which is equivalent to %@) a [O,a [i.e. Q! < c(B) = sup U(B), thus we have from (iv) sup,,x lim ~;~;(x)*c@)>cY or there exists x rX such that lim g(x)>cu. D

As an ~~plicati~~~ of this theorem we can give a proof of the fuzzy Tychc~no~ theorem [4] without use of the fuzzy Alexander’s subbase lemma.

roof. We useTheorem 5.1. (ii). Let LY < 1 and 3 E WV*+(fli,JXj) prime. Since 8 is prime

limg=adhg= inf G O&

= inf { Q! E 32;: cy closed in n Xi

jd 1

= inf 1 sup prT’(aj) E 3: (Y& closed in Xk, Jo finite subset of J iG3

Again since 3 is prime we have that out of each finite combination Supjaro pri’(aj)E 3 some p&u& 8. Thus

lim 3 2 inf {p$(oy ) E 8: QI closed in Xi9 j E 9).

NOW fcr any a closed in Xi and j E J put p = p$(ar), then

pri(fi)= prj(pr7'(cu))=of

and thus prj(p) = pr&), consequent!y

lim 8 23 inf {prir((pr&)): p E 8, j E 9)

= j$ pr;’ I

Lsf prj(&) 1

= inf pri’(ad rj(~))* jE.F

Now since 3 E *I#!: &) we have that

c”(B)= 4u;:‘. ;g*@)>O.

ut a?= c# +&“@,4, t” ten from the equivalence

v(x)>ct! - c”(~)~Y(x)>a’+~c*(~)

we have

and thus &VW$G be &). It is easily seen that then for all j E J

prj@) f t/k’“, (Xi),

and since Xi is fxxz; compact we have that for all j E J there exislts some xi E Xi for which

Putting x = (X&J %Me obtain

lim g(x) a tnf pr?‘(adh pry) icJ

tl inf adh prj(@)(xj) jd

If X and Y arz topological spaces and f : X + Y then f is continuous iff for all x E X and for all ult~a~~~er $I converging to x we have that f(%) converges to f(x), and also iff for any x E .X and any filter 9 such that x E adh 9 we have f(x)~ adh f(S). With the convergerxc theory we have built up we can show analogous results for fuzzy continuity.

MM& This ~011~~s easily from the definition of adh 8 and from the fact that f is fuzzy ~onti~~~~~~s iff for all lu, E I” jQF)B f(g) (see [S]). D

158 R. hwen / Convergence in fuzzy topological spaces

roof. The only if part follows trivially from the fact that if 8 is prime f(g) is also prime and from Proposition 4.1 (iii) and Theorem 6.1.

To show the if part we proceed as follows. Suppose that for any prefirter 8 E VW(X) it is true tha’t

(1)

then we have that

f(adh 3) s sup lim f(a). @M&&j

(2)

Now for all @ E SQB) we have that f@) is prime and thus there exists &’ E @,,,(f@)) ‘such that f(a) 3 @’ xf(g) and this implies that

sup lim f(@)S sup lim ($5’ @M%&H ~-‘ERVl(fGB)

= adh f(g). (3)

Combining (2) and (3) gives the desired result. Thus all that remains to be shown is (1).

is trivial. Tu show the other inequality we shall first make some preliminary remarks and notational conventions. Theorem 3.2 states that

9,,(g) = ((8, a): % ultra and compatible with 33.

We shall put

Y(8) = (94: ultra and compatible with 3).

Then

Since the unit interval I wit its us& crder is a complete chain it follows that Ix is completely distributive [ 11. LeL

Then we split each function q E @ into its component functions 9 =i (VI, cpzb so that

rpl(W E 6 and cp2(%) E %. Now applying complete distributivity on (4) gives

sup inf inf Fu= inf su %EY(‘) VE$j UE(u rPE@ QeSP(B)

cp1(%92(9c).

R. Lowen / Convergence in fuzzy topologacal spaces 159

ConsecJ lrently it now suffices to show that for all p E 45 there elrists same p E 8 such that

Suppose 1

Qy9Ta, Ql(*)~~(sr,~ fie

that for all v E 3 and for all finite subset &&‘j) of 9’(@) we have

(6)

then thks implies that

;JI_ PW) n v-‘IO, 11 f 0,

so that the family

U?

: f Y&(B) finite subset of P(R)]

Fo which by (7) is compatibie with 8. Consequently if q&) = $1 /]O, 11: v E B}, then there exists an ultrafilter % finer than SO u L&B). This imp! ses that 94 E 9’(# and that cpz( %) E %. But since SO c % also ~$94)’ E % which k zmpoesible.

This ~39110~s that for cp 6 @ there exists v E 3 and 9b(‘g) a finite subset of 9(B) such that

Let ~~($‘! and v be such and put

then obvtoasly p E 8 and since p s v we again have

Consequev : tly (6) is proved by

Combirk* : 4 ), (5) and (6) proves ahe missing inequality of (I). proof of I(:LL 3 theorem. 0

? r 4

160 R. Luwen / Convergence in fuzzy topdbgica~ qaces

El1 PI [31

[41

151

161

G. Birkhoff, Lattice Theory (AMS Colloquium publications, Vol XXV, 1967). N. Bourbaki, Topologie G&r&ale (Hermann, Paris, 1965). R. Lowen, Fuzzy topological spaces and fuzzy compactness, J. Math. Anal. and Appl. 56 (1976) 621-633. R. Lowen, Initial and final fuzzy topologies and the fuzzy TychonofI theorem, J. Math. Anal. and Appl. 58 (1977) 11-21. R.H. Warren, Neighborhoods, bases and continuity in fuzzy topological spaces, to appear in Rocky Mt. J. Math. LA. Zadeh, Fuzzy sets, Information and Control 8 (1965) 338-353.